Quantum feedback control: How to use verification theorems and viscosity solutions to find optimal protocols

Contemporary Physics

Volumn 49, Issue 6, 2008, Pages 435-448

  Jacobs, Kurt ^a   Shabani, Alireza ^b  

^a UNIVERSITY OF MASSACHUSETTS BOSTON   (United States)

^b University of Southern California ^*  (United States)

Author keywords

Quantum feedback control; Verification theorems; Viscosity solutions

Indexed keywords

EID: 62549098453     PISSN: 00107514     EISSN: 13665812     Source Type: Journal    
DOI: 10.1080/00107510802601781     Document Type: Article

References (90)

1. K. Jacobs and D.A. Steck, A straightforward introduction to continuous quantum measurement, Contemp. Phys. 47 (2006), pp. 279-303.
2. T. A. Brun, A simple model of quantum trajectories, Am. J. Phys. 70 (2002), pp. 719-737.
3. V.P. Belavkin, Non-demolition measurement and control in quantum dynamical systems, in Information, Complexity and Control in Quantum Physics, A. Blaquiere, S. Diner, and G. Lochak, eds., Springer-Verlag, New York, 1987.
4. V.P. Belavkin, Theory of the control of observable quantum systems, Automatica and Remote Control 44 (1983), pp. 178-188.
5. H.M. Wiseman, Adaptive phase measurements of optical modes: Going beyond the marginal Q distribution, Phys. Rev. Lett. 75 (1995), pp. 4587-4590.
6. H.M. Wiseman, Using feedback to eliminate back-action in quantum measurements, Phys. Rev. A 51 (1995), pp. 2459-2468.
7. M. Yanagisawa and H. Kimura, A control problem for Gaussian states, in Learning, Control and Hybrid Systems, Lecture Notes in Control and Information Sciences, Y. Yamamoto, and S. Hara, eds., Vol. 241, Springer-Verlag, New York, 1998, pp. 249-313.
8. A.C. Doherty and K. Jacobs, Feedback control of quantum systems using continious state estimation, Phys. Rev. A 60 (1999), pp. 2700-2711.
9. V.P. Belavkin, Measurement, filtering and control in quantum open dynamical systems, Rep. Math. Phys. 43 (1999), pp. 405-425.
10. A.C. Doherty, S. Habib, K. Jacobs, H. Mabuchi, and S.M. Tan, Quantum feedback control and classical control theory, Phys. Rev. A 62 (2000), 012105.
11. A.C. Doherty, K. Jacobs, and G. Jungman, Information, disturbance, and Hamiltonian quantum feedback control, Phys. Rev. A 63 (2001), 062306.
12. L.K. Thomsen, S. Mancini, and H.M. Wiseman, Spin squeezing via quantum feedback, Phys. Rev. A 65 (2002), 061801.
13. M. Yanagisawa and H. Kimura, Transfer function approach to quantum control-part I: Dynamics of quantum feedback systems, IEEE Trans. Automat. Contr. 48 (2003), pp. 2107-2120.
14. M. Yanagisawa and H. Kimura, Transfer function approach to quantum control Part II: Control concepts and applications, IEEE Trans. Automat. Contr. 48 (2003), pp. 2121-2132.
15. J. Geremia, J.K. Stockton, A.C. Doherty, and H. Mabuchi, Quantum kalman filtering and the heisenberg limit in atomic magnetometry, Phys. Rev. Lett. 91 (2003), 250801.
16. L. Bouten, M. Guta, and H. Maassen, Stochastic schrodinger equations, J. Phys. A 37 (2004), pp. 3189-3209.
17. M.R. James, Risk-sensitive optimal control of quantum systems, Phys. Rev. A 69 (2004), 032108.
18. L. Bouten, S. Edwards, and V.P. Belavkin, Bellman equations for optimal feedback control of qubit states, J. Phys. B 38 (2005), pp. 151-160.
19. R. van Handel, J.K. Stockton, and H. Mabuchi, Feedback control of quantum state reduction, IEEE Trans. Automat. Control 50 (2005), pp. 768-780.
20. J. Gough, V.P. Belavkin, and O.G. Smolyanov, Hamilton-Jacobi-Bellman equations for quantum filtering and control, J. Opt. B: Quantum Semiclass. Opt. 7 (2005), pp. S237-S244.
21. L. Bouten and R. van Handel, On the separation principle of quantum control, preprint (2008). arXiv: math-ph/0511021.
22. H.M. Wiseman and A.C. Doherty, Optimal unravellings for feedback control in linear quantum systems, Phys. Rev. Lett. 94 (2005), 070405.
23. M. Mirrahimi, P. Rouchon, and G. Turinici, Lyapunov control of bilinear Schrodinger equations, Automatica 41 (2005), pp. 1987-1994.
24. M. Mirrahimi, G. Turinici, and P. Rouchon, Reference trajectory tracking for locally designed coherent quantum control, J. Phys. Chem. A 109 (2005), pp. 2631-2637.
25. C. D'Helon and M.R. James, Stability, gain, and robustness in quantum feedback networks, Phys. Rev. A 73 (2006), 053803.
26. D.Y. Dong, C.L. Chen, Z.H. Chen, and C.B. Zhang, Estimation-based information acquisition in quantum feedback control, Dynam. Cont. Dis. Ser. B 13 (2006), pp. 1204-1208.
27. J. Combes and K. Jacobs, Rapid state-reduction of quantum systems using feedback control, Phys. Rev. Lett. 96 (2006), 010504.
28. C. Altafini, Homogeneous polynomical forms for simultaneous stabilizability of families of linear control systems: A tensor product approach, IEEE T. Automat. Contr. 51 (2006), pp. 1566-1571.
29. C. Altafini, Decoherence control by tracking a Hamiltonian reference molecule, Quant. Inf. Proc. 6 (2007), pp. 9-36.
30. C. Altafini, Feedback stabilization of isospectral control systems on complex flag manifolds: Application to quantum ensembles, IEEE T. Automat. Contr. 52 (2007), pp. 2019-2028.
31. F. Ticozzi and L. Viola, Single-bit feedback and quantum-dynamical decoupling, Phys. Rev. A 74 (2006), 052328.
32. M. Pavon and F. Ticozzi, On entropy production for controlled Markovian evolution, J. Math. Phys. 47 (2006), 063301.
33. M. Mirrahimi and R. van Handel, Stabilizing feedback controls for quantum systems, SIAM J. Control Optim. 46 (2007), pp. 445-467.
34. S.K. Wang, J.S. Jin, and X.Q. Li, Continuous weak measurement and feedback control of a solid-state charge qubit: A physical unravelling of non-Lindblad master equation, Phys. Rev. B 75 (2007), 155304.
35. K. Jacobs and A.P. Lund, Feedback control of nonlinear quantum systems: A rule of thumb, Phys. Rev. Lett. 99 (2007), 020501.
36. F. Ticozzi and L. Viola, Quantum Markovian subsystems: Invariance, attractivity, and control, IEEE T. Automat. Contr. 53 (2008), pp. 2048-2063.
37. J. Gough and M.R. James, The series product and its application to quantum feedforward and feedback networks, preprint (2007). arXiv:0707.0048.
38. M.R. James and J. Gough, Quantum dissipative systems and feedback control design by interconnection, preprint (2007). arXiv:0707.1074.
39. H.I. Nurdin, M.R. James, and I.R. Petersen, Coherent quantum LQJ control, preprint (2007). arXiv:0711,2551.
40. V.P. Belavkin, A. Negretti, and K. Molmer, Dynamical programming of continuously observed quantum systems, preprint (2008). arXiv:0805.4741.
41. W.P. Smith, J.E. Reiner, L.A. Orozco, S. Kuhr, and H.M. Wiseman, Capture and release of a conditional state of a cavity QED system by quantum feedback, Phys. Rev. Lett. 89 (2002), 133601.
42. M.A. Armen, J.K. Au, J.K. Stockton, A.C. Doherty, and H. Mabuchi, Adaptive homodyne measurement of optical phase, Phys. Rev. Lett. 89 (2002), 133602.
43. R.L. Cook, P.J. Martin, and J.M. Geremia, Optical coherent state discrimination using a closed-loop quantum measurement, Nature 446 (1992), pp. 774-777.
44. B.L. Higgins, D.W. Berry, S.D. Bartlett, H.M. Wiseman, and G.J. Pryde, Entanglement-free Heisenberg-limited phase estimation, Nature 450 (2007), pp. 393-396.
45. A. Naik, O. Buu, M.D. LaHaye, A.D. Armour, A.A. Clerk, M.P. Blencowe, and K.C. Schwab, Cooling a nanomechanical resonator with quantum back-action, Nature 443 (2006), pp. 193-196.
46. A.O. Niskanen, K. Harrabi, F. Yoshihara, Y. Nakamura, S. Lloyd, and J.S. Tsai, Quantum coherent tunable coupling of superconducting qubits, Science 316 (2007), pp. 723-726.
47. A.A. Houck, D.I. Schuster, J.M. Gambetta, J.A. Schreier, B.R. Johnson, J.M. Chow, L. Frunzio, J. Majer, M.H. Devoret, S.M. Girvin and R.J. Schoelkopf, Generating single microwave photons in a circuit, Nature 449 (2007), pp. 328-331.
48. D.I. Schuster, A.A. Houck, J.A. Schreier, A. Wallraff, J.M. Gambetta, A. Blais, L. Frunzio, J. Majer, B. Johnson, M.H. Devoret, S.M. Girvin and R.J. Schoelkopf, Resolving photon number states in a superconducting circuit, Nature 445 (2007), pp. 515-518.
49. J.H. Plantenberg, P.C. de Groot, C.J.P.M. Harmans, and J.E. Mooij, Demonstration of controlled-NOT quantum gates on a pair of superconducting quantum bits. Nature 447 (2007), pp. 836-839.
50. C. Ahn, A.C. Doherty, and A.J. Landahl, Continuous quantum error correction via quantum feedback control, Phys. Rev. A 65 (2002), 042301.
51. K. Jacobs, How to project qubits faster using quantum feedback, Phys. Rev. A. 67 (2003), 030301(R).
52. A. Hopkins, K. Jacobs, S. Habib, and K. Schwab, Feedback cooling of a nanomechanical resonator, Phys. Rev. B 68 (2003), 235328.
53. J.M. Geremia, Distinguishing between optical coherent states with imperfect detection, Phys. Rev. A 70 (2004), 062303.
54. J.F. Ralph, E.J. Griffith, T.D. Clark, and M.J. Everitt, Guidance and control in a Josephason charge qubit, Phys. Rev. B 70 (2004), 214521.
55. M. Sarovar, C. Ahn, K. Jacobs, and G.J. Milburn, Practical scheme for error control using feedback, Phys. Rev. A 69 (2004), 052324.
56. D. Steck, K. Jacobs, H. Mabuchi, T. Bhattacharya, and S. Habib, Quantum feedback control of atomic motion in an optical cavity, Phys. Rev. Lett. 92 (2004), 223004.
57. V. Steixner, P. Rabl, and P. Zoller, Quantum feedback cooling of a single trapped ion in front of a mirror, Phys. Rev. A 72 (2005), 043826.
58. N. Yamamoto, Parametrization of the feedback Hamiltonian realizing a pure steady state, Phys. Rev. A 72 (2005), 024104.
59. M. Yanagisawa, Quantum feedback control for deterministic entangled photon generation, Phys. Rev. Lett. 97 (2006), 190201.
60. H.M. Wiseman and J.F. Ralph, Reconsidering rapid qubit purification by feedback, New. J. Phys 8 (2006), p. 90.
61. P. Bushev, D. Rotter, A. Wilson, F. Dubin, C. Becher, J. Eschner, R. Blatt, V. Steixner, P. Rabl and P. Zoller, Feedback cooling of a single trapped ion, Phys. Rev. Lett. 96 (2006), 043003.
62. D.N. Matsukevich, T. Chaneliere, S.D. Jenkins, S.-Y. Lan, T.A.B. Kennedy, and A. Kuzmich, Deterministic single photons via conditional quantum evolution, Phys. Rev. Lett. 97 (2006), 013601.
63. L. Tian, Markovian feedback to control continuous variable entanglement, Phys. Rev. A 73 (2006), 010304(R).
64. G. Katz, M.A. Ratner, and R. Kosloff, Decoherence control by tracking a Hamiltonian reference molecule, Phys. Rev. Lett. 98 (2007), 203006.
65. L. Tian, Correcting low-frequency noise with continuous measurement, Phys. Rev. Lett. 98 (2007), 153602.
66. K. Jacobs, Feedback control for communication with nonorthogonal states, Quant. Information Comp. 7 (2007), pp. 127-138.
67. C. Hill and J.F. Ralph, Weak measurement and rapid state reduction in entangled bipartite quantum systems, New J. Phys 9 (2007), p. 151.
68. A.R.R. Carvalho and J.J. Hope, Stabilizing entanglement by quantum-jump-based feedback, Phys. Rev. A 76 (2007), 010301.
69. S.D. Wilson, A.R.P. Carvalho, J.J. Hope, and M.R. James, Effects of measurement backaction in the stabilization of Bose-Einstein condensate through feedback, Phys. Rev. A 76 (2007), 013610.
70. N. Yamamoto, K. Tsumura, and S. Hara, Feedback control of quantum entanglement in a two-spin system, Automatica 43 (2007), pp. 981-992.
71. B.A. Chase, A.J. Landahl, and J.M. Geremia, Efficient feedback controllers for continuous-time quantum error correction, Phys. Rev. A 77 (2008), 032304.
72. J. Combes, H.M. Wiseman, and K. Jacobs, Rapid measurement of quantum systems using feedback control, Phys. Rev. Lett. 100 (2008), 160503.
73. H.M. Wiseman and L. Bouten, Optimality of feedback control strategies for qubit purification, Quant. Inform. Proc. 7 (2008), pp. 71-83.
74. T. Sagawa and M. Ueda, Second law of thermodynamics with discrete quantum feedback control, Phys. Rev. Lett. 100 (2008), 080403.
75. C.A. Fuchs and K. Jacobs, Information-tradeoff relations for finite-strength quantum measurements, Phys. Rev. A 63 (2001), 062305.
76. W.H. Fleming and R.W. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag, New York, 1975.
77. V.G. Boltyanskii, Sufficient conditions for optimality and the justification of the dynamic programming method, SIAM J. Control 4 (1966), pp. 326-361.
78. R.B. Vinter and R.M. Lewis, Verification theorem which provides a necessary and sufficient condition for optimality, IEEE T. Automat. Contr. 25 (1980), pp. 84-89.
79. X.Y. Zhou, J. Yong, and X. Li, Stochastic verification theorems within the framework of viscosity solutions, SIAM J. Cont. Optim. 35 (1997), pp. 243-253.
80. F. Gozzi, A. Swiech, and X.Y. Zhou, A corrected proof of the stochastic verification theorem within the. framework of viscosity solutions, SIAM J. Cont. Optim. 43 (2005), pp. 2009-2019.
81. M.G. Crandall, H. Ishii, and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, B. Am. Math. Soc. 27 (1992), pp. 1-67.
82. A. Shabani and K. Jacobs, Locally optimal control of quantum systems with strong feedback, Phys. Rev. Lett. 101 (2008), 230403.
83. D.T. Gillespie, The mathematics of Brownian motion and Johnson noise, Am. J. Phys. 64 (1996), pp. 225-240.
84. K. Jacobs, Topics in quantum measurement and quantum noise, Ph.D. thesis, Imperial College, London, 1998.
85. C.A. Fuchs, Distinguishability and accessible information in quantum theory, PhD thesis, University of New Mexico (UNM, Albuquerque), 1995.
86. C.A. Fuchs, Information gain vs. state disturbance in quantum theory, Fort. der Phys. 46 (1998), pp. 535-565.
87. V. Vedral, M.B. Plenio, K. Jacobs, and P.L. Knight, Statistical inference, distinguishability of quantum states, and quantum entanglement, Phys. Rev. A 56 (1997), pp. 4452-4455.
88. P. Diţǎ, Factorization of unitary matrices, J. Phys. A 36 (2003), pp. 2781-2789.
89. P. Whittle, Optimal Control, John Wiley & Sons, Chichester, 1996.
90. P.S. May beck, Stochastic Models, Estimation and Control, Vol. I and II, Academic Press, New York, 1982.